Representations of surface groups in the projective general linear group
arXiv:0901.2314
Abstract
Given a closed, oriented surface X of genus g>1, and a semisimple Lie group G, let R_G be the moduli space of reductive representations of the fundamental group of X in G. We determine the number of connected components of R_PGL(n,R), for n>=4 even. In order to have a first division of connected components, we first classify real projective bundles over such a surface. Then we achieve our goal, using holomorphic methods through the theory of Higgs bundles over compact Riemann surfaces. We also show that the complement of the Hitchin component in R_SL(3,R) is homotopically equivalent to R_SO(3).
v3: included an erratum (Section 12) which shows why Theorem 1.3 (stating that the Hitchin component in R_SL(3,R) is homotopically equivalent to R_SO(3)) is not correct, even though the original manuscript is left unchanged. This erratum has been published in Int. J. Math., 30, No. 2 (2019) 1992001